Adaptability and phenotypic stability of common bean genotypes through Bayesian inference

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Adaptability and phenotypic stability of

common bean genotypes through Bayesian

inference

A.M. Corrêa1_{, P.E. Teodoro}2_{, M.C. Gonçalves}3_{, L.M.A. Barroso}4_, M. Nascimento4_{, A. Santos}5_{and F.E. Torres}1

1_{Departamento de Fitotecnia, Universidade Estadual do Mato Grosso do Sul,} Aquidauana, MS, Brasil

2_{Departamento de Biologia Geral, Universidade Federal de Viçosa, Viçosa,} MG, Brasil

3_{Departamento de Estatística, Universidade Federal da Grande Dourados,} Dourados, MS, Brasil

4_{Departamento de Estatística, Universidade Federal de Viçosa, Viçosa, MG,} Brasil

5_{Departamento de Melhoramento Genético,}

Universidade Estadual do Norte Fluminense Darcy Ribeiro, Campo dos Goytacazes, RJ, Brasil

Corresponding author: P.E. Teodoro E-mail: [email protected] Genet. Mol. Res. 15 (2): gmr.15028260 Received December 11, 2015

Accepted January 18, 2016 Published April 26, 2016

DOI http://dx.doi.org/10.4238/gmr.15028260

ABSTRACT. This study used Bayesian inference to investigate the genotype x environment interaction in common bean grown in Mato Grosso do Sul State, and it also evaluated the efficiency of using informative and minimally informative a priori distributions. Six trials were conducted in randomized blocks, and the grain yield of 13 common bean genotypes was assessed. To represent the minimally informative a priori distributions, a probability distribution with high variance was used, and a meta-analysis concept was adopted to represent the

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informative a priori distributions. Bayes factors were used to conduct comparisons between the a priori distributions. The Bayesian inference was effective for the selection of upright common bean genotypes with high adaptability and phenotypic stability using the Eberhart and Russell method. Bayes factors indicated that the use of informative a priori distributions provided more accurate results than minimally informative a priori distributions. According to Bayesian inference, the EMGOPA-201, BAMBUÍ, CNF 4999, CNF 4129 A 54, and CNFv 8025 genotypes had specific adaptability to favorable environments, while the IAPAR 14 and IAC CARIOCA ETE genotypes had specific adaptability to unfavorable environments.

Key words: Phaseolus vulgaris L.; Bayes factor; Informative prior; Genotype x environment interaction

INTRODUCTION

Brazil is the largest producer of common bean (Phaseolus vulgaris L.), with a production estimated at 3.0 million tons in 2014 (FAO, 2015) and average productivity of 0.94 Mg/ha (CONAB, 2015). However, this productivity is low considering its productive potential, which can reach 4500 kg/ha. There are several reasons for this issue, such as the inappropriate use of genotypes that are not adapted to the various growing regions.

A major challenge for plant breeders is determining the appropriate common bean genotypes for the various regions of Brazil due to genotype x environment (GE) interactions, which determine the differential response of genotypes among environments. To reduce the effects of GE interactions, it is convenient to know their magnitude and to identify more stable genotypes adapted to specific environments (Cruz and Regazzi, 2007). In this context, several methods to study adaptability and stability have been used to measure GE interactions in common bean (Coimbra et al., 1999; Carbonell et al., 2004; Ribeiro et al., 2009; Pereira et al., 2009, 2011; Torga et al., 2013), predominantly based on linear regression models (Eberhart and Russell, 1966) and multivariate analyses, such as additive main effects and multiplicative interaction analysis (AMMI) (Gauch, 2006).

Traditional methods that predict genotype performances in multiple environments are based on a “classic” (or frequentist) approach to statistics, which estimates one or more parameters from a set of observations. The variables are used to make statements about a statistical model, which is characterized in terms of parameters with a “true” value (Couto et al., 2015). An alternative to classical statistical inference is Bayesian inference, which is based on the principles of likelihood and the incorporation of a priori information (Silva et al., 2013).

This approach uses three basic concepts: initial information (a priori probability), generally assumed as a joint probability law on the parameters before obtaining the private information of the samples y₁, ..., y_n of the random variable; the probabilistic model of random response variable y, with which it obtains the likelihood of the sample; and Bayes’ theorem, which combines the a priori information and the likelihood function and generates a posterior distribution (Cotes et al., 2006; Silva et al., 2013). Nascimento et al. (2011) formulated a Bayesian approach using the Eberhart and Russell (1966) method to evaluate GE interaction, and concluded that the selection of genotypes in different environments was most accurate

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when a priori information was used. It is a robust statistical procedure with many possible applications, but its use in plant genetic improvement is still limited (Teodoro et al., 2015).

Thus, the aim of this study was to use Bayesian inference to investigate the GE interaction in common bean grown in Mato Grosso do Sul State, and we also evaluated the efficiency of using informative and minimally informative a priori distributions.

MATERIAL AND METHODS

Six trials (environments) were conducted from 2002 to 2006. The environments were composed of a combination of location, growing season, and agricultural year (Table 1). Thirteen genotypes (Rudá, Aporé, Xamego, Pérola, Ouro Negro, Diamante Negro, IAPAR 14, EMGOPA 201, and IAC CARIOCA ETE and the lines BAMBUÍ, CNF 4999, CNF 4129 A 54, and CNFv. 8025) were evaluated in a randomized block design with three replications. Experimental units consisted of four 1.5 m long rows of plants spaced 0.50 m apart. Grain yield was evaluated in the central rows, being expressed in Mg ha-1_{, considering the area}

harvested and adjusting the data for 13% moisture.

1_{According to Köppen-Geiger classification.}

Table 1. Environments, site, climate, latitude, longitude, altitude, harvest, and agricultural year of trials

conducted in the State of Mato Grosso do Sul.

Environment Site Climate1 _Latitude _Longitude _{Altitude (m)} _Harvest _{Agricultural year}

1 Dourados Cwa 22°12' 54°48' 452 Rainy 2000/2001

2 Dourados Cwa 22°12' 54°48' 452 Drought 2000/2001

3 Aquidauana Aw 20°20' 55°48' 207 Drought 2000/2001

Data were subjected to individual ANOVA, and fixed and other treatment effects were considered to be random. The results indicated that the relationship between the highest and lowest mean square of the individual ANOVA of the residues did not exceed a 7:1 ratio, which allowed the implementation of the joint analysis of trials (Banzatto and Kronka, 2006). The data were subsequently subjected to adaptability and stability analyses using the Eberhart and Russell (1966) method.

The adopted linear regression model of Eberhart and Russell (1966) was:

where Y_ij is the observed mean of genotype i in environment j; β_0i is the linear coefficient related to i-th genotype; β_li is the regression coefficient of genotype i; I_j is the environmental index j; and Ψ_ij represents the random errors that are compounded by the regression deviation of (Equation 1)

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genotype i in environment j and by the mean error associated with the mean. The environmental index was estimated using the following equation:

where Y_m is the overall mean; Yj is the mean of environment j; and n is the number of

environments.

According to the Eberhart and Russell (1966) method, genotype adaptability was measured using the parameter β_1i, while the stability of behavior was evaluated using the variance of the regression deviations (σ2

di) and the coefficient of determination (R2), which

is an auxiliary measure for stability assessment (Cruz and Regazzi, 2007). The R2_value

indicates acceptable predictability when σ2

di is significant and R2 is higher than 80%. Using

this frequentist approach, the hypotheses of interest are: H₀:β_li = 1 versus H₁:β_li ≠ 1 and H₀: σ2 di

= 0 versus H₁: σ2

di > 0, which are assessed using t and F statistics, respectively.

For the Bayesian analysis, we considered all genotypes that were evaluated in Corrêa et al. (2009), which were used as references for the specification of a priori distributions (Table 2).

(Equation 2)

Table 2. Estimates and means (β_0i) associated with adaptability (β_1i), stability (σ2

di), and the coefficient of

determination (R2_{), which were obtained using the Eberhart and Russell method (1966), as described in Corrêa}

et al. (2009).

Genotype 0i (Mg/ha) 1i 2di R2

RUDÁ 2.18 0.97 128,693 67.33 APORÉ 2.31 0.75 110,006 58.78 XAMEGO 2.04 0.93 95,113 71.24 PÉROLA 1.71 0.82 87,035 67.51 OURO NEGRO 1.96 1.32 212,871 70.59 DIAMANTE NEGRO 1.75 0.99 36,570 85.75 IAPAR 14 1.87 0.69 151,370 47.49 EMGOPA-201 2.31 1.13 270,922 58.40

IAC CARIOCA ETE 1.73 0.61 121,449 45.89

BAMBUÍ 1.63 1.31 393,732 57.04

RIO TIBAGI 1.76 1.17 96,833 79.43

CNF 4129 A 54 1.36 1.06 199,921 62.04

CNFv 8025 1.97 1.24 245,919 65.17

With the Bayesian approach, we considered the following statistical model, developed by Nascimento et al. (2011):

where each Y_ij observation was assumed to have the following distribution:

where the likelihood function for each genotype i is given by:

(Equation 3)

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To estimate the adaptability and stability parameters, a priori distributions were assigned for the parameters. The following distributions were considered for β_0i, β_li, and σ2

di:

The equal mean and variance for the reverse gamma values were calculated as follows, respectively:

When we assumed independence between the parameters of these distributions, the joint a priori distributions for each genotype were given by:

To make inferences about the parameter of interest, the marginal a posteriori distributions must be obtained. When we denoted the parameter vectors for each genotype i using the following equation:

(where P = 1, 2, 3), the marginal a posteriori distribution for the parameter θ_pi was obtained using the following integral:

(Equation 5) (Equation 6) (Equation 7) (Equation 8) (Equation 9) (Equation 10) (Equation 11) (Equation 12)

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(i.e., the integral in relation to all vector parameters with the exception of the p-th component). In most cases, these integrals were complex and did not have exact solutions. To work around this problem, we used other methodologies. For example, we obtained a sample of the joint a posteriori distribution using the Markov chain and Monte Carlo (MCMC) method, which was used to determine the moments associated with the marginal distributions of interest (Cassela and George, 1992). In this study, the analyses were performed using R software (R Development Core Team, 2015) with codes developed by Nascimento et al. (2011), and the joint distribution sample was obtained using the MCMC regression function of the MCMC package.

To evaluate the influence of a priori information when estimating the adaptability and stability parameters, we utilized two different models: informative a priori distributions and minimally informative a priori distributions (Nascimento et al., 2011). In Model 1, the informative a priori information was derived from the application of meta-analysis techniques, which were characterized using information from the study by Corrêa et al. (2009).

The 13 genotypes evaluated in the trials were used as references for a priori specification. Therefore, all of the genotypes presented in Table 2 were considered for Bayesian analysis. Information was inserted into the analysis using the assumed values for a priori distribution parameters (i.e., hyperparameters). These values were based on the mean and variance values of the samples that were composed using the parameter estimates obtained from the cited references (Table 2), which resulted in the following distributions:

where represents the means of the β_0i estimates; represents the means of the β_li estimates; Var( ) is the variance of the mean values; Var( ) is the variance of the mean values; and α_i and b_i represent values obtained using the following system resolution equations:

(Equation 13) (Equation 14) (Equation 15) (Equation 16) (Equation 17) (Equation 18)

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In Model 2, minimally informative a priori distributions were used, and these distributions represented probability distributions with large variance. The following distributions were adopted:

The comparisons between Models 1 and 2 (i.e., between informative and minimally informative a priori distributions) were performed using the Bayes factor calculation (Kass and Raftery, 1995), which was conducted using the Bayes Factor function of the MCMC package. According to Jeffreys (1961), Bayes factors can be interpreted as follows: FBij < 1 provides evidence in favor of model j; 1 ≤ FBij < 3 provides moderate evidence in favor of model i; 3 ≤ FBij < 10 indicates substantial evidence in favor of model i; 10 ≤ FBij < 30 demonstrates strong evidence in favor of model i; 30 ≤ FBij < 100 provides very strong evidence in favor of model 1; and FBij ³100 indicates decisive evidence in favor of model i.

Regarding the stability parameter (σ2

di), samples of its marginal distributions were

obtained indirectly, because this parameter represents a function. When values for σ2 i were

obtained indirectly in each iteration, we acquired values for σ2

di using the following expression:

where MSR is the mean square of the residue provided by ANOVA, and r is the number of repetitions in the trial.

The hypotheses of interest were tested by constructing credibility ranges for the parameters, and the intervals were obtained directly from the marginal a posteriori distribution of the parameters. Thus, the credibility interval (CI) for θ_i, with a probability of covering δ, is given by: (Equation 19) (Equation 20) (Equation 21) (Equation 22) (Equation 23) (Equation 24)

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where θ_* and θ* represent the lower and upper limits of the CI, respectively. Since the Gibbs sampler is an iterative algorithm, it is necessary to check its convergence. In this study, the convergence was assessed by applying the Heidelberger and Welch (1983), Raftery and Lewis (1992), and Geweke (1992) criteria, which were implemented in the Bayesian Output Analysis package (BOA) of the R program (R Development Core Team, 2015).

Regarding Bayesian analyses of adaptability and stability for each parameter of the adopted regression model, 110,000 iterations in the Gibbs sampler algorithm with a warm period (“burn-in”) of 10,000 iterations were considered. To obtain a non-correlated sample, we considered the spacing between the sampling points of two iterations (“thinning”), which resulted in a final sample size of 50,000. The samples represented samples of marginal a posteriori distributions for each parameter under which the inference was conducted (Nascimento et al., 2011).

RESULTS AND DISCUSSION

In joint analyses, all of the effects were significant (P ≤ 0.01) (Table 3), which indicated contrasts between the environments and the occurrence of genotypic differential responses to environmental effects. This could be confirmed by examining differences in the soil and climatic features of each environment, including altitude, latitude, longitude, climate type, soil type, rainfall, and temperature (Table 1). Similar results were obtained by Coimbra et al. (1999), Carbonell et al. (2004), Ribeiro et al. (2009), Pereira et al. (2009, 2011), and Torga et al. (2013) in previous studies, which found significant differences associated with the effects of genotypes, environments, and GE interactions when evaluating common genotypes in multi-environment trials in Brazil. The existence of significant GE interactions for grain yield indicated that the stability and adaptability analyses were suitable, and this result was further supported by the fact that edaphoclimatic factors had the greatest influence on the adaptability and stability of common bean genotypes.

(Equation 25)

*Significant at a 0.01 probability level using the F test.

Table 3. Summary of joint ANOVA for grain yield (Mg/ha) of 13 common bean genotypes that were evaluated

in six environments in the State of Mato Grosso do Sul.

Sources of variation d.f. Mean square

Blocks/Environment 10 28,717.86 Genotypes 12 1,362,132.53* Environment 5 3,182,552.19* Genotypes × Environment 60 303,445.22* Error 146 22,387.42 Mean (Mg/ha) - 1.89 Coefficient of variation (%) - 10.96

Regarding the convergence for all simulated chains, the dependency factor of Raftery and Lewis (1992) gave values lower than five, and p-values based on the criteria of Geweke (1992) were higher than the prefixed significance level (P ≤ 0.05) (i.e., both criteria

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indicated convergence of the chains generated by the Gibbs sampler). In order to confirm this convergence, we also used the criteria of Heidelberger and Welch (1983), which determined whether the chain values were derived from a stationary distribution.

The adaptability and stability parameter estimated values were obtained by calculating the a posteriori means, which are presented together with their respective CIs in Table 4. In Model 1 (informative a priori), of the 13 genotypes of common bean evaluated by the Bayesian approach, the EMGOPA-201, BAMBUÍ, CNF 4999, CNF 4129 A 54, and CNFv 8025 genotypes were classified as having specific adaptability to favorable environments (LI β_1i > 1). The IAPAR 14 and IAC CARIOCA ETE genotypes presented specific adaptability to unfavorable environments (LS β_1i < 1). The other genotypes were classified as having general adaptability and stability, since the value of 1 fell within a CI of 95%. However, when considering the analysis of genotypes under Model 2 conditions (minimally informative a priori), all genotypes of common bean showed specific adaptability to unfavorable environments (LI β_1i < 1). Therefore, it is possible to infer that the use of the frequentist model, in which a priori information is not taken into account, tends to conclude that the genotypes of common bean exhibit stability and adaptability to different environments, but this does not favor reliable genotype recommendations. Nascimento et al. (2011), Couto et al. (2015), and Teodoro et al. (2015) obtained similar results after evaluating the adaptability and phenotypic stability of alfalfa, popcorn, and cowpea genotypes, respectively.

Table 4. Estimates of a posteriori mean (β0i, in Mg/ha) and credibility intervals (95%) of the adaptability (β1i) and stability (σ_i2) parameters for common bean genotypes when considering informative and minimally informative a priori distributions.

Genotype LI0i 0i_LS0i LI1i 1i LS1i i2x1000 LIdi2x1000 di2x1000 LSdi2x1000 Informative a priori RUDÁ 2.03 2.18 2.33 0.68 0.94 1.20 35.94 12.05 29.01 60.09 APORÉ 2.08 2.21 2.34 0.34 0.57 0.82 27.12 7.31 20.21 44.04 XAMEGO 1.74 1.86 1.99 0.65 0.88 1.11 23.97 5.69 17.06 37.92 PÉROLA 1.56 1.69 1.82 0.58 0.82 1.06 28.02 7.84 21.11 45.38 OURO NEGRO 1.62 1.75 1.88 0.89 1.13 1.38 27.69 7.59 20.78 45.23 DIAMANTE NEGRO 1.60 1.68 1.77 0.83 0.99 1.16 10.81 -1.24 3.90 13.36 IAPAR 14 1.81 2.03 2.24 0.20 0.53 0.87 80.97 35.69 74.06 144.29 EMGOPA -201 1.85 1.97 2.10 1.05 1.29 1.51 23.96 5.63 17.05 37.93

IAC CARIOCA ETE 1.64 1.70 1.76 0.22 0.34 0.47 5.29 -4.15 -1.62 3.04

BAMBUÍ 1.91 2.07 2.22 1.19 1.47 1.73 39.55 13.74 32.64 67.34

CNF 4999 1.64 1.75 1.87 1.21 1.44 1.65 20.59 3.81 13.68 31.81

CNF 4129 A 54 1.29 1.47 1.64 1.04 1.35 1.64 51.11 19.54 44.19 89.61

CNFv 8025 2.03 2.14 2.25 1.01 1.22 1.43 19.60 3.39 12.69 29.75

Minimally informative a priori

RUDÁ 1.70 2.17 2.62 -0.05 0.93 1.91 32.84 52.32 32.15 134.41 APORÉ 1.79 2.19 2.58 -0.34 0.51 1.35 24.47 37.17 23.78 99.81 XAMEGO 1.47 1.85 2.22 0.06 0.86 165 21.85 32.46 21.16 88.94 PÉROLA 1.27 1.68 2.08 -0.04 0.82 1.68 25.59 39.25 24.89 104.21 OURO NEGRO 1.32 1.73 2.12 0.21 1.06 1.91 24.89 37.99 24.19 101.41 DIAMANTE NEGRO 1.42 1.68 1.93 0.46 0.99 1.53 9.97 10.99 9.28 40.08 IAPAR 14 1.34 2.03 2.69 -1.10 0.35 1.79 71.93 12.39 71.24 296.01 EMGOPA -201 1.57 1.95 2.31 0.55 1.34 2.13 21.61 32.03 20.92 87.97

IAC CARIOCA ETE 1.52 1.69 1.87 -0.05 0.32 0.70 49.73 20.00 42.81 195.89

BAMBUÍ 1.62 2.10 2.56 0.54 1.55 2.56 35.20 56.62 34.51 144.21

CNF 4999 1.40 1.75 2.09 0.78 1.51 2.24 18.44 26.31 17.75 74.85

CNF 4129 A 54 9.30 1.47 1.99 0.42 1.55 2.67 43.94 72.75 43.24 179.61

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By comparing the estimates of the parameters obtained using the two models, similar differences in magnitude between the parameters were observed. Thus, it is necessary to determine which of the two models exhibited a higher quality setting, and this answer was provided using Bayes factor calculations (Nascimento et al., 2011). All of the genotypes evaluated exhibited Bayes factors that were greater than 41 (Table 5), resembling the results obtained by Nascimento et al. (2011), Couto et al. (2015), and Teodoro et al. (2015).

Table 5. Values obtained for Bayes factors (BF) of models compared using informative a priori (i) and

minimally informative a priori (j) distributions for the common bean genotypes evaluated.

Genotype BF_ij RUDÁ 55.80 APORÉ 53.90 XAMEGO 52.80 PÉROLA 53.90 OURO NEGRO 54.40 DIAMANTE NEGRO 46.20 IAPAR 14 63.10 EMGOPA -201 53.60

IAC CARIOCA ETE 41.10

BAMBUÍ 58.30 CNF 4999 52.00 CNF 4129 A 54 59.80 CNFv 8025 51.10 RUDÁ 55.80 APORÉ 53.90 XAMEGO 52.80

These results indicated that the use of informative a priori distributions provided more accurate results. Even though the a priori information for each genotype was based on a single study, similar results were found in studies that evaluated the adaptability and phenotypic stability of the dry matter yield of alfalfa genotypes (Nascimento et al., 2011), popcorn grain yield (Couto et al., 2015), and cowpea grain yield (Teodoro et al., 2015; Barroso et al., 2016). Therefore, it is expected that the results will be more precise in studies examining the stability and adaptability of other crops with more available information for meta-analysis.

The Bayesian inference, in conjunction with the Eberhart and Russell method (1966), was an effective method for the selection of common bean genotypes with high adaptability and phenotypic stability. Moreover, Bayes factors indicated that the use of informative a priori distributions provided more accurate results than the use of minimally a priori distributions. According to Bayesian inference, the EMGOPA-201, BAMBUÍ, CNF 4999, CNF 4129 A 54, and CNFv 8025 genotypes had specific adaptability to favorable environments, while the IAPAR 14 and IAC CARIOCA ETE genotypes presented specific adaptability to unfavorable environments.

Conflicts of interest

The authors declare no conflict of interest.

ACKNOWLEDGMENTS

We thank the Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES) and the National Council for Scientific and Technological Development (CNPq) for financial support.

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